Weak invariance principle for finite-populationU-statistics

Under the weakest possible conditions, we establish the weak invariance principle for finite-populationU-statistics in this paper. It is worth while to point out that, for the sampling without replacement, the sequence of random delements inC[0, 1], associated with the sample partial sums or theU-statistics, converges in law to the standard Brown bridge, but not to the Brown motion as in the usual case of replacement sampling.     

Normal approximation for a random elliptic equation

We consider solutions of an elliptic partial differential equation in \(\mathbb{R }^d\) with a stationary, random conductivity coefficient that is also periodic with period \(L\) . Boundary conditions on a square domain of width \(L\) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit \(L \rightarrow \infty \) , this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size \(L\) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.     

Normal approximation of random permanents

For random permanents, we obtain an estimate of the rate of convergence in the central limit theorem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 922–927, July, 1995.     

The not-quite non-atomic game: Normal approximation

Measure games with a large number of players are frequently approximated by nonatomic games. In fact, however, while it is true that values of large measure games will, under certain reasonable circumstances, converge to the value of a non-atomic game, it is also true that this convergence is quite slow. Using the multi-linear extension and the central limit theorem, we obtain an approximation which (because it is based on the normal distribution) we call the normal approximation. We show that, for two examples with several hundred and several thousand players respectively, the normal approximation is much better than the non-atomic approximation.     

Normal cone approximation and offset shape isotopy

This work addresses the problem of the approximation of the normals of the offsets of general compact sets in Euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the μ-reach of the compact set, a recently introduced notion of feature size. As a consequence, we provide a sampling condition that is sufficient to ensure the correctness up to isotopy of a reconstruction given by an offset of the sampling. We also provide a notion of normal cone to general compact sets that is stable under perturbation.     

Normal approximation for linear stochastic processes and random fields in Hilbert space

The central limit theorem is proved for linear random fields defined on an integer-valued lattice of arbitrary dimension and taking values in Hilbert space. It is shown that the conditions in the central limit theorem are optimal. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 421–428, September, 2000.     

Normal approximation of a functional of a Gaussian field

The asymptotic normality of a functional of a strongly correlated Gaussian random field having the meaning of the random volume of a bounded realization of the field over a parallelepiped or triangle is established in this paper. In addition some problems of geometric probabilities are solved: the distribution density of the distance between independent random vectors having uniform distribution in a rectangle or triangle is found.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 55–60, 1988.     

Normal correlation coefficient of non-normal variables using piece-wise linear approximation

The correlation coefficient of non-normal variables is expressed as a function of the correlation coefficient of normal variables using piece-wise linear approximation of each univariate transform of normal to anything, and the second order moments of a multiply truncated bivariate normal distribution. For the inverse problem, an algorithm iterates this analytic function in order to assign a normal correlation coefficient to two non-normal variables. The algorithm is applied for the generation of randomized bivariate samples with given correlation coefficient and marginal distributions and used in a randomization test for bivariate nonlinearity. The test correctly does not reject the null hypothesis of linear correlation if the nonlinearity is plausible and due to the sample transform alone.     

Normal spectral approximation inC *-algebras and in von Neumann-algebras

In this paper the general problem of normal spectral approximation in a unitalC ^*-algebraA is studied. If Δ is a closed and convex subset of the complex plane then the distance from a normal elementa ofA to the setN _A (Δ) of all normal elements with spectrum in Δ equals the one-sided Hausdorff distance from σ(a) to Δ. As a consequence every normal elementa has anN _A (Δ)-approximant which is a function ofa. Furthermore the approximant is unique whenever every point of σ (a) has the same distance to Δ. These results are shown for the approximation in theC ^*-norm as well as in another topological equivalent norm.

     

Poisson approximation for large deviations

Upper and lower bounds are given for P(S ≤ k), 0 ≤ k ≤ ES, where S is a sum of indicator variables with a special structure, which appears, for example, in subgraph counts in random graphs. in typical cases, these bounds are close to the corresponding probabilities for a Poisson distribution with the same mean as S. There are no corresponding general bounds for P(S ≥ k), k > ES, but some partial results are given.     

An algorithm for segment approximation

Summary An algorithm for computing a set of knots which is optimal for the segment approximation problem is developed. The method yields a sequence of real numbers which converges to the minimal deviation and a corresponding sequence of knot sets. This sequence splits into at most two subsequences which converge to leveled sets of knots. Such knot sets are optimal. Numerical results concerning piecewise polynomial approximation are given.     

Topological grammars for data approximation

A method of topological grammars is proposed for multidimensional data approximation. For data with complex topology we define a principal cubic complex of low dimension and given complexity that gives the best approximation for the dataset. This complex is a generalization of linear and non-linear principal manifolds and includes them as particular cases. The problem of optimal principal complex construction is transformed into a series of minimization problems for quadratic functionals. These quadratic functionals have a physically transparent interpretation in terms of elastic energy. For the energy computation, the whole complex is represented as a system of nodes and springs. Topologically, the principal complex is a product of one-dimensional continuums (represented by graphs), and the grammars describe how these continuums transform during the process of optimal complex construction. This factorization of the whole process onto one-dimensional transformations using minimization of quadratic energy functionals allows us to construct efficient algorithms.     

Newton's method for fractal approximation

The problem of fitting a given function in theL ^q norm with a function generated by an iterated function system can be rapidly solved by applying Newton's method on the parameter space of the iterated function system. The key to this is a method for calculating the derivatives of a potential function with respect to the parameters.     

Diffusion approximation for nonparametric autoregression

A nonparametric statistical model of small diffusion type is compared with its discretization by a stochastic Euler difference scheme. It is shown that the discrete and continuous models are asymptotically equivalent in the sense of Le Cam's deficiency distance for statistical experiments, when the discretization step decreases with the noise intensity ε. Received: 12 April 1996 / Revised version: 29 October 1997     

Crank-nicolsen-Galerkin approximation for Maxwell's equations

The Maxwell equations are formulated as an evolution equation in a suitable chosen Hilbert space involving a densely defined closed skew-Hermitian operator which generates a unitary group. A Crank-Nicolsen-Galerkin approximation is then established and convergence is shown by arguments from the theory of approximation of groups of operators.     

Diffusion approximation for storage processes

For a sequence of storage processes with general release rate functions which contain, as a special case, queuing processes, we show that under appropriate conditions, suitably normalized processes for storage processes converge to diffusions in the sense of law.